Robert S. Ball

“ The geologist, who has hitherto guided our view, cannot render us much further assistance; but the physicist is at hand—he teaches us that the warm globe on which life is beginning has passed in its previous stages through every phase of warmth, of fervour, of glowing heat, of incandescence, and of actual fusion; and thus at last our retrospect reaches to that particular period of our earth's past history which is specially illustrated by the modern doctrine of Time and Tide. ”

“ You will go to the pier head, or, better still, to some place where the rise and fall of the mere waves of the sea will not embarrass you in your work. You must note by your watch the time when the tide is highest. ”

“ Not to embarrass ourselves with the perplexities of a problem so complicated as our solar system is in its entirety, we shall for the sake of clear reasoning assume an ideal system, consisting of a sun and [156] a large planet—in fact, such as our own system would be if we could withdraw from it all other bodies, leaving the sun and Jupiter only remaining. We shall suppose, of course, that the sun is much larger than the planet, in fact, it will be convenient to keep in mind the relative masses of the sun and Jupiter, the weight of the planet being less than one-thousandth part of the sun. ”

“ The earth is incessantly losing heat; and though in a day, a week, or a year the loss may not be very significant, yet when we come to deal with periods of time that have to be reckoned by millions of years, it may well be that the effect of a small loss of heat per annum can, in the course of these ages, reach unimagined dimensions. Suppose, for instance, that the earth experienced a fall of temperature in its interior which amounted to only one-thousandth of a degree in a year. ”

“ So far as a human life or the life of the human race is concerned, the decline of a tenth of a degree per century in the earth's internal heat is absolutely void of significance. I cannot, however, impress upon you too strongly, that the mere few thousands of years with which human history is cognizant are an inappreciable moment in comparison with those unmeasured millions of years which geology opens out to us, or with those far more majestic periods which the astronomer demands for the events he has to describe. ”

“ The orbit of the moon might have a much smaller radius than it has at present, provided that the speed was sufficiently increased to compensate for the increased attraction which the earth would exercise at the lessened distance. Indeed, I am here only stating what every one is familiar with under the form of Kepler's Law, that the square of the periodic time is in proportion to the cube of the mean distance. To each distance of the moon therefore belongs an appropriate speed. The energy due to the moon's position and the energy due to its motion are therefore connected together. ”

“ There is indeed another way of demonstrating that a condition of the system in which the day has assumed equality with the month must necessarily be one of dynamical equilibrium. We have shown that the energy which the tides demand is derived not from the mere fact that there are high tides and low tides, but from the circumstance that these tides do rise and fall; that in falling and [125] rising they do produce currents; and it is these currents which generate the friction by which the earth's velocity is slowly abated, its energy wasted, and no doubt ultimately dissipated as heat. ”

“ They are always doing work of some kind, whether that be merely heating the particles of water by friction, or vaguely transporting sand from one part of the ocean to the other. Useful work or useless work are alike for the purpose of my argument. We know that work can never be done unless by the consumption or transformation of energy. For each unit of work that is done—whether by any machine or contrivance, by the muscles of man or any other animal, by the winds, the waves, or the tides, or in any other way [48] whatever—a certain equivalent quantity of energy must have been expended. ”

“ I may here note a point about which people sometimes find a little difficulty. The moon constantly turns the same face towards the earth, and therefore people are sometimes apt to think that the moon performs no rotation whatever around its own axis. But this is indeed not the case. The true inference to be drawn from the constant face of the moon is, that the velocity of rotation about its own axis is equal to that of its rotation around the earth; in fact, the moon revolves around the earth in twenty-seven days, and its rotation about its axis is performed in twenty-seven days also. ”

“ Of the two endowments of energy and of moment of momentum originally conferred on our system the moment of momentum is the entailed estate. No matter how the bodies may move, no matter how their actions may interfere with one another, no matter how this body is pulled one way and the other body that way, the conservation of moment of momentum is not imperilled, nor, no matter what losses of heat may be experienced by radiation, could the store of moment of momentum be affected. ”

“ It actually happens that though the moon does cause the tides, yet when those tides consume energy they draw it not from the distant moon, but from the vast supply which they find ready to their hand, stored up in the rotation of the earth. ”

“ We have here an illustration of the contrast between stable and unstable equilibrium. It will be found generally that a body is in equilibrium when its centre of gravity is at its highest point or at its lowest point; there is, however, this important difference, that when the centre of gravity is highest the equilibrium is unstable, and when the centre of gravity is lowest the equilibrium is stable. The potential energy of an egg poised on its end in unstable equilibrium is greater than when it lies on its side in stable equilibrium. ”

“ There cannot be a doubt that in this epoch of the earth's history, so long as the price of coal is only a few shillings a ton, the tide-mill, even though we seem to get its power without current expense, is vastly more expensive than a steam-engine. Indeed, Sir William Thomson remarks, that wherever a suitable tidal basin could be found, it would be nearly as easy to reclaim the land altogether from the sea. ”

“ I do not know that there is anything impossible in the supposition that the sun might, by an augmented volume and an augmented velocity of rotation, contain many times the moment of momentum that it has at this moment. It therefore follows that if it had happened that Jupiter constantly bent the same face to the sun, there would apparently be nothing impossible in the fact that Jupiter had been born of the sun, just as the moon was born of the earth. These same considerations should also lead us to observe with still more special attention the development of the earth-moon system. ”

“ The usual way in which the powers of the mathematician have been awakened into action is by the effort to remove some glaring discrepancy between an imperfect theory and the facts of observation. The genius of a Laplace or a Lagrange was expended, and worthily expended, in efforts to show how one planet acted on another planet, and produced irregularities in its orbit; the genius of an Adams and a Leverrier was nobly applied to explain the irregularities in the motion of Uranus, and to discover a cause of those irregularities in the unseen Neptune. ”

“ It is almost unavoidable for us to make a surmise as to the cause by which the moon had come into this remarkable position close to the earth at the most critical epoch of earth-moon history.
With reference to this Professor Darwin has offered an explanation, which seems so exceedingly plausible that it is impossible to resist the notion that it must be correct. I will ask you to think of the earth not as a solid body covered largely with ocean, but as a glowing globe of molten material. In a globe of this kind it is possible for great undulations to be set up. ”

“ The circumstance that heat is a form of energy indeed negatives this supposition. For our system possesses energy of all the different kinds: there is energy due to the motions both of rotation and of revolution; there is energy due to the fact that the mutually attracting bodies of our system are separated by distances of enormous magnitude; and there is also energy in the form of heat; and the laws of heat permit that this form of energy shall be radiated off into space, and thus disappear entirely, in so far as our system is concerned. ”

“ At all events, the illustration may suffice to show, that the fact that we are not able to prove by our instruments that the earth is cooling is no argument whatever against the inevitable law, that the earth, like every other heated body, must be tending towards a lower temperature.
Without pretending to any numerical accuracy, we can at all events give a qualitative if not a quantitative analysis of the past history of our earth, in so far as its changes of temperature are concerned. A million years ago our earth doubtless contained appreciably more heat than it does at present. ”

“ The tide curve is thus rapidly drawn out; and so expeditious is the machine, that the tides of a port for an entire year can be completely worked out in a couple of hours.
While the student or the philosopher who seeks to render any account of the tide on dynamical grounds is greatly embarrassed by the difficulties introduced by friction, we, for our present purpose [38] in the study of the great romance of modern science opened up to us by the theory of the tides, have to welcome friction as the agent which gives to the tides their significance from our point of view. ”

“ It is the consideration of these points that forms the stronghold of the doctrine of tidal evolution. For when we find that a theory depending upon influences that undoubtedly exist, and are in ceaseless action around us, can at the same time bring into connection and offer a common explanation of a number of phenomena which would otherwise have no common bond of union, it is impossible to refuse to believe that such a theory does actually correspond to nature. ”

“ We might as well try to get a steam-engine which would produce enough heat by friction not only to supply its own boilers, but to satisfy all the thermal wants of the whole parish. We must therefore adopt the other alternative. The tides do not draw their energy from the moon; they draw it from the store possessed by the earth in virtue of its rotation. ”

“ We may for simplicity regard the moon as a point, and as this attracts each particle of the earth, the total effect of the moon on the earth may be represented by a single force. By the law of equality of action and reaction, the force of the earth on the moon is to be represented by an equal and opposite force. If there were no tides then the moon's force would of course pass through the earth's centre ”

“ On no other supposition can we reasonably account for the fact that our barometric records fail to afford us any very distinct evidence as to the existence of tides in the atmosphere. For you will, of course, remember that our atmosphere may be regarded as a deep and vast ocean of air, which embraces the whole earth, extending far above the loftiest summits of the mountains.
It is one of the profoundest of nature's laws that wherever friction takes place, energy has to be consumed. ”

“ It is sometimes possible to effect the desired end by applying the power of the tide. A sea-wall judiciously thrown out will sometimes concentrate the tide into a much narrower channel. Its daily oscillations will be accomplished with greater vehemence, and as the tide rushes furiously backwards and forwards over the obstacle, the incessant action will gradually remove it, and the impediment to navigation may be cleared away. Here we actually see the tides performing a piece of definite and very laborious work, to accomplish which by the more ordinary agents would be a stupendous task. ”

“ A mechanical device, however ingenious may be the construction, or however accurate the workmanship, can never possess what is called perpetual motion. It is needless to enter into details of any proposed contrivance of wheels, of pumps, of pulleys; it is sufficient to say that nothing in the shape of mechanism can work without friction, that friction produces heat, that heat is a form of energy, and that to replace the energy consumed in producing the heat there must be some source from which the machine is replenished if its motion is to be continued indefinitely. ”

“ I have already shown that the fact of the ebbing and the flowing of the tide necessitates an expenditure of energy, and we saw that this energy must come either from that stored up in the earth by its rotation, or from that possessed by the moon in virtue of its distance and revolution. The law of the conservation of spin will enable us to decide at once as to whence the tides get their energy. Suppose they took it from the moon, the moon would then lose in energy, and consequently come nearer the earth. ”

“ But the most important constituent of the store of energy in the earth-moon system is that contributed by the earth itself. I do not now speak of the energy due to the velocity of the earth in its orbit round the sun. The moon indeed participates in this equally with the earth, but it does not affect those mutual actions between the earth and moon with which we are at present concerned. ”

“ If I might venture on a very homely illustration, I may say that the moon, like a troublesome fellow, is constantly annoying the earth by dragging its waters backward and forward by means of tides; and the earth, to free itself from this irritating interference, tries to push off the aggressor and to make him move further away. ”

“ For simplicity we may make the assumption that the whole of the earth is buried beneath the ocean, and that the moon is placed in the plane of the equator. We may also entirely neglect for the present the tides produced by the sun, and we shall also make the further assumption that friction is absent. What friction is capable of doing we shall, however, refer to later on. The moon will act on the ocean and deform it, so that there will be high tide along one meridian, and high tide also on the opposite meridian. ”